Student Sorting and Implications for Grade Inflation1

Michael C. Herron2 Zachary D. Markovich3

September 27, 2016

1The authors thank Craig Volden for the conversation that inspired this paper and Mark McPeek, Rus-sell Muirhead, two anonymous referees, and seminar participants at Dartmouth College, the University ofCalifornia at San Diego, the University of Virginia, and Yale University for helpful comments.

2Visiting Scholar, Hertie School of Governance, Berlin, Germany, and Professor of Government, Dart-mouth College. 6108 Silsby Hall, Hanover, NH 03755-3547 (michael.c.herron@dartmouth.edu).

3Research associate, Harvard Business School (zmarkovich@hbs.edu).

Abstract

There is a sizable literature on higher education, both in the United States and beyond, that draws

attention to the phenomenon known as grade inflation. We offer an interpretation of grade inflation

that turns on the choices students have over academic departments, and we argue that patterns in

grades cannot be considered in isolation from the incentives that students have to sort themselves

strategically across departments. Our argument draws on a game-theoretic model in which students

of varying abilities face a choice between enrolling in a department whose grades are inflated and

thus ability-concealing versus enrolling in a department whose grades are ability-revealing. In

equilibrium, all grades are high. Nonetheless, what appears to be grade inflation is a result of the

fact that the ability-revealing department in our model attracts highly talented students seeking to

distinguish themselves from students of lesser ability, who avoid said department because enrolling

in it is costly. Our formalization shows how student sorting can confound grades, and it implies

that a full understanding of university’s grade distribution requires knowing which departments in

the university are ability-concealing and which, in contrast, are ability-revealing.

Introduction

Grade inflation appears by many measures to be endemic across the higher education landscape.

Ray (2014) documents rising grades in the United Kingdom, and as of 2012, “80 percent of stu-

dents [in Germany] graduate[d] with one of the top two grades.”1 A 2010 study of grade trends at

over 160 American universities revealed that mean grade point average has risen by nearly a tenth

of a point per decade (Rojstaczer and Healy, 2010), and the revelation that the most commonly

awarded grade at Harvard University is an “A” unleashed a firestorm of media attention on the

university’s grading policies.2 In 2001, 91 percent of Harvard’s graduating class received honors,

rendering this indicator of student ability essentially meaningless in the sense of distinguishing

among Harvard undergraduates.3 And at Yale University, approximately 62 percent of grades were

“A” and “A-” as of Spring, 2012, whereas in 1963 this proportion was only ten.4 Empirical re-

search on grading tends to show that the phenomenon of grade inflation is broad-based although

1On the German case, see “Grade inflation blows up German universities,” The Local.de, available athttp://www.thelocal.de/20121219/46847 (last accessed September 19, 2016).

2For press coverage of grade inflation in American higher education, see “Doesn’t Anybody Get a C Any-more?” The Boston Globe, October 5, 2008, available at http://www.boston.com/bostonglobe/magazine/articles/2008/10/05/doesnt_anybody_get_a_c_anymore (last accessedFebruary 15, 2015); “How to Fix College Grade Inflation,” U.S. News & World Report, December 26, 2013,available at http://www.usnews.com/opinion/blogs/economic-intelligence/2013/12/26/why-college-grade-inflation-is-a-real-problem-and-how-to-fix-it(last accessed February 14, 2015); “Harvard professor says grade inflation rampant,” TheBoston Globe, December 44, 2013, available at http://www.bostonglobe.com/metro/2013/12/03/harvard-professor-raises-concerns-about-grade-inflation/McZHfRZ2RxpoP5Xvwged1N/story.html (last accessed February 14, 2015); and,“The Most Commonly Awarded Grade at Harvard Is an A,” The Atlantic, December 4,2013, available at http://www.theatlantic.com/education/archive/2013/12/the-most-commonly-awarded-grade-at-harvard-is-an-a/282020/ (last accessedApril 28, 2015).

3See “Harvard’s dirty, little secret is out – grade inflation / Graduating with honors is abreeze,” Boston Globe, October 14, 2001, available at http://www.sfgate.com/news/article/Harvard-s-dirty-little-secret-is-out-grade-2868775.php (last access April 23,2015).

4See “Grade expectations,” Yale Alumni Magazine, September/October 2013, available at https://www.yalealumnimagazine.com/articles/3735/grade-expectations (last accessedApril 20, 2015).

1

Kuh and Hu (1999) argue that, in the United States, grade inflation is disproportionately a feature

of research universities and elite liberal arts colleges.

Grade inflation can be associated with numerous social problems. Because inflated grades

can mask variance in student abilities (Sabot and Wakeman-Linn, 1991; Pattison, Grodsky and

Muller, 2013), post-education institutions that must contend with universally high grades will have

difficulty selecting the best students for the most demanding tasks. This will impose an efficiency

cost on society. In addition, differing rates of grade inflation across departments can skew student

enrollment decision; STEM (science, technology, engineering, and mathematics) departments are

disproportionately less likely to grade inflate, and a student who receives an early and poor grade

in a STEM class is less likely to major in a STEM field (Sabot and Wakeman-Linn, 1991; Strenta

et al., 1994; Ost, 2010). Moreover, there are gender and socioeconomic differences in the types

of students whose grades have increased over time; women in particular are disproportionately

likely to enroll in disciplines prone to grade inflation (Riegle-Crumb and King, 2010; Riegle-

Crumb et al., 2012) and in disciplines perceived as “soft” (Carnevale, Smith and Melton, 2011;

Swift et al., 2013). This will make it difficult for the most talented women to stand out from the

rest. There is also evidence that, among American high school students, grades have increased

more slowly for Hispanic students than members of other ethnic groups (Sawtell, Kimmel and

Scheuneman, 2003), and in general minority students consistently receive lower grades than white

students (Farkas and Hotchkiss, 1989; Van Laar et al., 1999). We return later to a discussion of

social consequences of grade inflation, but for the moment it suffices to note that differential rates

of grade inflation across department and student types can reinforce stereotypes which affect some

students more than others.

As we review shortly, scholars have offered a variety of explanations for observed upward

trends in grades. Some explanations for grade inflation focus more heavily on (and tend to blame)

the role of educational institutions and the incentives that their rules and procedures engender

while others draw attention to the behaviors of students and the extent to which their choices over

2

departments and courses have downstream effects on measures like average grades. Certainly there

are connections between these two ways of thinking—students make decisions in an environment

crafted by universities—but it is nonetheless still useful to distinguish between incentives facing

faculty members and incentives facing students.

Having noted this dichotomy, we offer an interpretation of grade inflation that turns on student

choice over departments or, more broadly, courses of study. In particular, we present a formal

model that allows students to choose whether they want to study in a department whose grades

are inflated versus a departments whose grades are accurate. Inflated grades can be thought of

as ability-concealing and accurate grades as ability-revealing. The students in the model differ

in their underlying levels of talent, and this induces students to have different preferences over

the extent to which they want their grades to reveal their underlying abilities. Students in the

model sort themselves strategically across departments, and this confounds the interpretation of

grades. Among other things, the model shows that trends in grades—upward or downward—can

be induced by changes in student sorting abilities. In light of this, caution should be exercised

when drawing connections between trends in grades and potentially deleterious consequences for

educational institutions and society at large.

In what follows, we discuss literature on grade inflation, and we then present our model and

explain its various components. The basic model has two types of equilibria, and we show that, in

equilibrium, strategic sorting by students leads to high grades. We then present an extension of the

model which generalizes our initial, and coarse, characterization of student ability and also allows

for what we call an education bonus. In the extension, we observe high grades in equilibrium, and

the extension yields a somewhat counter-intuitive result: the more difficult a university’s ability-

revealing department, the higher are average grades and the more it appears that grade-inflation is

ubiquitous. This extension has policy implications for efforts to rein in high grades, and we discuss

such implications and others in the conclusion.

3

Grade inflation

Concerns over grade inflation in higher education are not new (e.g., Ekstrom and Villegas, 1994),

and as we noted in the introduction one consequence of grade inflation is the masking of true stu-

dent abilities. To the point, Sabot and Wakeman-Linn (1991) show that introductory course grades

received in low-grading departments are better predictors of student performance in future classes

compared to grades given in what appear to be grade-inflating departments. Similarly, they show

that alternate predictors of student ability (e.g., standardized test scores, parental education levels,

high school grades, and so forth) are associated with student grades in low-grading departments

but not departments that routinely give high grades. These dual findings show that inflated grades

effectively mask student abilities and diminish the extent to which grades signaling underlying

skills and talent levels.5

One of the commonly cited reasons for an increase in grades across American colleges and

universities has been the increasing weight placed on instructor evaluations in hiring and tenure

decisions (Stratton, Myers and King, 1994; Eiszler, 2002). Students tend to give better evalua-

tions to professors who award them higher grades (Johnson, 2003), and thus an increased reliance

on teacher evaluations during evaluation processes can create incentives for high grades. Nelson

and Lynch (1984) argue that the relationship between evaluations and grades can be exacerbated

by stagnating faculty salaries, and Pressman (2007) notes that pressure for high grades will tend

to be stronger for untenured but tenure-track professors and strongest for adjunct faculty, whose

employment depends on enrollments.

Perrin (1998) and Kelly (2009) draw attention to the fact that a university professor may, in the

course of grading, compare her students not just to other students at her own university but also to

the typical American student. Perrin writes, “[Professors] imagine our students at a mythical Aver-

5Millman et al. (1983) argue that grade inflation through 1983 had not in the United States eliminated thesignaling value of grades, and Pattison, Grodsky and Muller (2013) makes similar claims with contemporarydata.

4

age U., and give the grades they would get there.” If a faculty member believes that her institution’s

admissions policies lead to a highly-talented student body, then it follows that said faculty mem-

ber should in general assign high grades. On this point, see Achen and Courant (2009) and their

anecdote of “a [University of Michigan] chemistry professor who had stuck to the standards of his

own undergraduate work for decades, but who came to notice that incoming graduate students at

Michigan often had better grades than graduates of [his] department with similar knowledge and

skill.”

Other arguments on the subject of grade inflation focus specifically on student course selection.

For example, in some American universities students are allowed to take classes without grades

appearing on transcripts. Strategically-minded students may seek to take advantage of this practice

by ensuring that grades for their most difficult classes are not visible. If students strategically select

certain classes to have non-visible grades, then average grade point averages may increase even in

the face of fixed grading policies (Birnbaum, 1977). Foreshadowing the model that we present

here, Prather, Smith and Kodras (1979) argue that changes in average grades can reflect changes in

enrollment patterns; their empirical research finds that “English majors tend to receive relatively

higher grades in education courses than in their other courses, while the grades they receive for

physical science and foreign language courses are, on the average, lower. Physical science courses

generally record lower grades for all majors, while teacher education courses comparatively record

higher grades for all majors” (pp. 21-22). The implication of such a finding is that average grades

reflect student selection into coursework of interest.

Bar, Kadiyali and Zussman (2009) analyze data from Cornell University, and find that publicly-

available median grades allow students to select into leniently graded classes. Strenta et al. (1994)

and Ost (2010) have similar findings, and both note that low grades in first year science classes

increases the probability that a student chooses a non-science major. There is also evidence that

grading policies respond to the perceived value of a major (Freeman, 1999). When a department’s

graduates do not perform as well on the job market, they are forced to “buy” students with higher

5

grades. Jewell, McPherson and Tieslau (2013) validate this, finding that there is substantial depart-

mental variation in grade inflation.

In contrast to theories of grade inflation that consider student and institution incentives, Adel-

man (2008) argues that increasing grades may be explained by improved student ability and/or

teaching quality. In this view, increasing grades are not inherently problematic. Brighouse (2008)

emphasizes that, to assume that there has been no improvement in student quality over the past 30

years is to assume that have been no efficiency gains in higher education over this period.

There is some formal work on grade inflation, but the literature is not extensive. Four examples

are Yang and Yip (2003), Chan, Hao and Suen (2007), Franz (2010), and Popov and Bernhardt

(2013). In the former, schools have incentives to give high grades because this helps weaker

students obtain jobs; this leads to labor market inefficiencies. In Chan, Hao and Suen, employers

cannot determine whether students with high grades are high quality or whether the university that

granted said grades is an easy-grading institution; as in Yang and Yip, this leads to inefficient labor

market outcomes. Franz (2010) models professor-student interactions with an eye on the costs on

faculty that students impose by requesting high grades. In equilibrium, the “nuisance” students

in Franz lead professors to inflate grades. Finally, Popov and Bernhardt (2013) propose a model

where universities compete for job market outcomes. They show that more selective universities

have the strongest incentives to grade inflate.

Students and institutions in this limited formal literature are strategic, but extant models in the

literature do not allow students to sort themselves across departments (or other academic units) in

the way described here. Insofar as contemporary university students appear to be very attuned to

grading policies and how they vary by field of study (and even by class and professor), our model

of sorting fills a gap in the literature.

6

Model

We now describe a model that sheds light on the dynamics of student sorting across university

departments and resulting patterns in grades. The model is set in a single university and includes

a group of students and two departments. Its premise is that the students in the university have

already been admitted but must choose a department or course of study in which to enroll. As will

be clear, a student’s choice between departments is informed by her interest, or lack thereof, in

signaling her intrinsic ability level to a labor market that she will enter upon graduation.

Students

In the model there are two types of students, low ability and high ability. A student’s ability is

fixed and exogenous, and let the proportion of high ability students in our hypothetical university

be πH ∈ (0, 1). All students know their own abilities, either low or high, but face the problem of

credibly signaling their talents to a labor market.6 In the sense of Spence (1973), students attend

university in order to earn a transcript which can subsequently be used to indicate ability. Like the

students in Love and Kotchen (2010), our students value grades because of what they signal to a

labor market.

We assume that the post-university labor market rewards high ability, and by implication being

high ability is valuable to a student. In particular, if a student is known to be of high ability, then

after her education is complete she earns a wage that we call wH . In contrast, a student known to be

of low ability earns in a post-education market a wage of wL where wH > wL.7 If after graduation

a student’s ability is not known to the labor market, then said student receives a wage proportional

to the probability that she is of high ability. For example, if the market believes that a student is of

6A plausible way for students to know their own abilities is through introductory “weed-out” classes thatare highly discriminatory and do not requirement commitment to a major or course of study. One could alsoargue that students know their abilities via secondary education coursework, but this might be troubling assecondary education may suffer from the same sorts of grading dynamics that are present in universities.

7This parallels an assumption made by Popov and Bernhardt (2013).

7

high ability with probability 1/4, then her post-education wage will be 1/4 × wH + 3/4 × wL.

We do not model the post-education labor market explicitly, and implicit in our assumptions about

post-education wages is that firms in the labor market are risk neutral and that all students have

jobs after graduating.

This latter assumption is not binding; one could treat what we call the low wage wL as an

unemployment or welfare benefit. Moreover, one could also interpret what we call the high post-

education wage wH as a placement in a post-graduate institution like a medical school or law

school. As long as wH > wL, which we assume throughout, being of high ability is better than

being of low ability, ceteris paribus.

All of our results that follow depend on the difference between wH and wL, and this means that

we can simplify our presentation without any loss of generality. With this in mind, we henceforth

assume that wH = 1 and wL = 0. This normalization is costless and reduces some of the algebra

in our forthcoming equilibria.

Asserting that there are two types of students—low and high ability—is a simplification. We

could have assumed that student ability exists on a continuum, and we consider such an extension

to our model after we present initial results.

Departments

We assume that our hypothetical university contains departments whose grades are either ability-

revealing or ability-concealing. These types of departments differ only in the manner in which

they assign grades to their students. It is broadly accepted that grading practices differ between

departments and that these differences impact student enrollment patterns (Sabot and Wakeman-

Linn, 1991; Strenta et al., 1994; Bar, Kadiyali and Zussman, 2009; Ost, 2010).

An ability-revealing department is one that offers courses with regular and discriminating ex-

aminations, projects, assignments, and so forth. These examinations, say, allow the department to

know whether a given student enrolled in the department is of low ability or is of high ability, and

8

the department indicates this knowledge via grades. In particular, a grade-revealing department

assigns an “A” grade to high ability students (because these students did well on the department’s

examinations) and a “B” grade to low ability students. Assuming that an ability-revealing de-

partment assigns grades of “A” and “B,” as opposed to “A” and “C” or “A” and “D,” is of no

consequence. The key here is that an ability-revealing department assigns grades that discriminate

between low and high ability students. The regular and discriminating examinations given in an

ability-revealing department require an effort cost for enrolled students, who know that these ex-

aminations and related assignments will ascertain their underlying abilities. Let c denote the effort

cost associated with enrolling in an ability-revealing department, and we assume that c > 0. Recall

that high ability students identified as such earn a normalized wage of one post-graduation. If this

wage of one is interpreted as the present value of a stream of wages, then c is presumably much

smaller than one and in particular should be understood as being quite close to zero.

In contrast, an ability-concealing department is one whose courses do not discriminate between

low and high ability students. The courses in an ability-concealing department are by definition

not excessively challenging, and the key is that all students enrolled in them receive excellent

grades, in particular, marks of “A.” Moreover, the lack of discriminating examinations means that

students in an ability-concealing department are not subject to the effort cost comparable to the

cost c incurred by students in a grade-revealing department.8

We could have assumed that students enrolled in an ability-concealing department are forced

to pay an effort cost akin to the cost required of students in a grade-revealing department. Had

we done this, our model’s equilibria, which follow shortly, would have been a function of the

difference between the effort cost required of a student in an ability-revealing department and the

cost required of a student in a grade-concealing department. Thus, the assumption that c > 0 is

akin to assuming that it is more costly for a student to enroll in an ability-revealing department

8The key here is that an ability-concealing department gives all enrolled students “A” grades. One couldposit that the examinations in such a department are ability-revealing but that the faculty in said departmentsimply ignore this when assigning grades.

9

than in an ability-concealing department.

We also could have assumed that low and high ability students pay different costs for attending

an ability-revealing department. That is, we could have posited that high ability students pay cH

when enrolling in an ability-revealing department and low ability students cL. It would probably

be natural to assume that cL > cH > 0, meaning, low ability students have to work harder in

ability-revealing departments than do high ability students. Regardless, our main results do not

depend on whether costs for attending an ability-revealing department vary by student type.9

Labor market

As we noted above, we do not formally model firms in a post-education labor market, nor do we

model, say, admissions committees in graduate institutions. However, we assume that the market

knows which types of departments are ability-revealing and which are ability-concealing. This

does not strike us as a particularly strong assumption although we recognize that one could argue

that firms, graduate schools, and other post-graduate institutions are not informed about which

departments in universities give ability-discriminating grades.

Average grades and grade inflation

As will be clear shortly, our model generates a distribution of grades across students, and from

this distribution we can calculate average grades in equilibrium. Before we discuss equilibria, we

9Our model assumes that a student’s being of low or high ability is a generic statement about underlyingtalent. However, some academic departments, civil engineering for example, teach specific skills that ahigh ability student who focuses in a non-engineering area will not have, despite being of high ability. Aslong as the types of departments that teach specific skills are ability-revealing, then our model applies tothem. Departments to which our model might not apply are those that teach specific skills yet are ability-concealing. If testing for the presence of skills requires careful examinations, it is hard to imagine that manydepartments with specific skill requirements—engineering, medicine, and so forth—are like this. Finally,it is worth emphasizing that there may be differences in grading policies between departments that teachsimilar skills and provide similar credentials. For example, electrical engineering and computer sciencemight have tracks that qualify students for similar technical jobs but nonetheless that have different gradingpractices.

10

need to define grade inflation. In our two-type model, high ability students cannot receive inflated

grades because they are of high ability; only low ability students can have inflated grades. Thus,

we say that a low-ability student receives an inflated grade if said student receives a grade that is

equal to or greater than the grade received by a high ability student. If a low ability student receives

a grade of “A,” then we say that this student’s grade is inflated. Why? A high ability student who

attends an ability-revealing department will by definition receive an “A” grade. Thus, a low ability

student who also receives an “A” grade has an inflated grade.

It is important to distinguish between the fraction of inflated grades in equilibrium and the

fraction of high grades. These two quantities are distinct: a high grade is not necessarily inflated if

it is earned by a high ability student. Distinguishing between inflated versus high grades helps us

understand the difference between situations where there is serious grade inflation (many students

receive grades greater than their abilities) versus situation where there are many high yet accurate

grades.

Beyond grade inflation

Key to our model is the opportunity for individuals—in our case, students—to signal their types

by engaging in costly behavior—taking classes in an ability-revealing department. What makes

grading particularly interesting is the fact that grading scales typically have upper bounds.10 As we

will see in our forthcoming equilibrium analysis, this can lead to situations where many students

receive the same top grade even if said students differ in their underlying abilities.

If grades lacked an upper bound, then there would be no such thing as the maximum grade a

student could receive. Still, an ability-concealing department in this situation could nonetheless

give all of its students identical marks. This would be fully consistent with the department’s being

ability-concealing. In other words, in an ability-concealing department, the distribution of grades

10Another feature of grades is that they usually are discrete. However, our model does not have anythingto say about this feature of grading.

11

can be compressed even the absence of something like a top grade. With this in mind, even if

grades were in theory unbounded, our model still would speak to the existence of different types

of departments, some that offer grades that discriminate between students and others that do not.

Many evaluation systems have “top” grades, however, and our model thus applies to situations

beyond university grading. Firms regularly have to assess their employees, for example. Some

tasks in a firm are presumably ability-concealing and others are ability-revealing. If a firm’s em-

ployees have a voice in choosing what they do within a firm, then the dynamics we have touched on

in our model might lead to low-quality employees choosing tasks or career paths that are ability-

concealing; high-quality employees will do the opposite.

Similarly, suppose that a set of legislators in a city council are of two types, low and high

quality, and are faced with a policy reform problem, i.e., how best to reform a municipal social

welfare program. A legislator can choose between an easy reform, one that does not accomplish

much but poses little risk, or a challenging reform, one that might expose a low quality legislator

to the critique that he or she did not design a reform very well. Suppose that after a reform effort,

an interest group generates a rating—that is, a grade—of said reform. In this framework, high-

quality legislators may seek to distinguish themselves by choosing challenging reforms if voters

are sufficiently attuned to the interest group’s post-reform ratings. However, if interest group

ratings are bounded above like grades, then the notion of rating inflation is plausible if, say, all

legislators choose easy reform projects that shed little light on underlying legislator quality.

Our general point here is that the study of grading extends beyond academic environments. The

model we have offered is one of evaluation in an environment in which tasks differ in the extent

to which they identify talented individuals. The equilibrium dynamics that are forthcoming below

will be helpful guides in understanding all such environments.

12

Equilibria

We assume that a university has two departments, one that is ability-revealing and the other, ability-

concealing; the assumption of two departments is not constraining and we comment on it later. The

university’s students, 1−πH of whom are of low ability and πH of whom are of high ability, choose

simultaneously whether to enroll in one of two departments. Subsequent to this the departments

assign grades, and then the students enter a post-education labor market. As we have noted above,

the labor market pays each student based on her ability or her expected ability if the market cannot

discern based on grades whether the student is of low or high ability. Only the students in our

model are strategic actors, and a student’s utility is equal to her post-education wage minus the

effort cost c if the student enrolled in an ability-revealing department.

The essence of a game-theoretic model is that the utility of a player selecting a given strat-

egy is conditional on the strategies chosen by other players. One can see evidence of this type

of mutual interdependence in the model described here when considering the value of a student’s

enrolling in an ability-concealing department. To make this clear, suppose that a low ability stu-

dent is considering such a department. If all other students also enroll in the ability-concealing

department, and if there are mainly high ability students because πH is large, then our hypothetical

low ability student will be pleased with her choice. Why? She will be grouped with both low and

high ability students, and the post-education labor market will not know if she is low ability. If,

though, high ability students all enroll in the ability-revealing department, then by enrolling in the

ability-concealing department she is signalling that she is of low ability. This will be less pleasing

as the post-education labor market will know for certain that she is of low ability. This example,

which foreshadows the pooling and separating equilibria that we discuss below, shows why the

value to any student (in this example, a low ability student, but the same sort of example applies to

high ability students as well) of enrolling in an ability-concealing department is conditional on the

choices made by others.

13

Figure 1: Model equilibria as a function of πH

0 1πH

1− c

Separation Pooling

The (Bayesian) equilibria of the model depend on the fraction πH of high ability students

and the cost c of enrolling in an ability-revealing department. As characterized in Lemma 1 and

described visually in Figure 1, the model always has a separating equilibrium and has a pooling

equilibrium if πH ≥ 1− c.

Lemma 1. There is always a separating equilibrium wherein low ability students enroll in an

ability-concealing department and high ability students enroll in an ability-revealing department.

If πH ≥ 1 − c, then in addition there is a pooling equilibrium wherein all students enroll in an

ability-concealing department. Regardless of the values of πH and c, there is never a pooling

equilibrium in which all students enroll in an ability-revealing department.

The proof of Lemma 1 is in the appendix, and the lemma characterizes the model’s pooling

and separating equilibria. We discuss these equilibria in this order.

Pooling equilibrium. When πH ≥ 1− c, the model has a pooling equilibrium in which all stu-

dents enroll in an ability-concealing department. Consider the implication of the aforementioned

(weak) inequality. When it holds, then the fraction πH of high ability students is large and indeed

almost all students are of high ability. The intuition for this is as follows. Recall that c denotes an

effort cost, presumably short-term, that students enrolling in an ability-revealing department pay as

a consequence of having to endure discriminating examinations, assignments, and so forth. Recall

as well that high ability students identified as such earn a normalized wage of one post-graduation.

Since this wage of one is best interpreted as the present value of a stream of wages, then c is pre-

sumably much smaller than one and in particular should be understood as being quite close to zero.

And, when c is close to zero, then 1− c is close to one. Thus, πH ≥ 1− c implies that we have a

rarefied situation in which there are almost no low ability students at all.

14

Continuing, when every student enrolls in the ability-concealing department, which is what

happens in the pooling equilibrium under consideration, then all students receive the same grades,

in particular grades of “A.” An outside observer assessing this situation—in which the fraction of

“A” grades is one and there is no variance in awarded grades—might be inclined to say that this is

a situation characterized by rampant grade inflation. Such a characterization would be inaccurate,

however. Rather, the fraction of inflated grades in the pooling equilibrium is 1 − πH , which is

very small and actually quite close to zero given our earlier discussion of πH and c. Intuitively,

what our hypothetical observer is seeing is not a consequence of low ability students flocking to an

easy-grading department in order to hide their low ability levels, a pattern of behaviors that would

in fact indeed induce high rates of grade inflation. Rather, the situation the observer sees is one in

which high ability students flock to such a department.

When πH ≥ 1 − c, it would be hard to argue that grade inflation as we have defined it is

troublesome. After all, when this inequality holds, almost every student is of high ability, and

thus the fact that almost every student receives an “A” grade does not connote a serious mismatch

between grades and underlying abilities. There is of course some mismatch here, as 1− πH of the

student body—the low ability students—receives grades that are inflated. However, 1 − πH must

be quite small for the pooling equilibrium to exist.

One might want to argue that the presence of all students’ pooling on an ability-concealing

department is an observable indication that almost every student is of high ability. Thinking em-

pirically about actual trends in grades, this reverses the concern that many have articulated about

inflated transcripts. To the point, in the pooling equilibrium discussed here, an abundance of stu-

dents who enroll in an ability-concealing department means that almost every student is highly

talented and not, say, that all students lack ability and are choosing an ability-concealing depart-

ment because they fear being exposed as such by a grade-revealing department. In our pooling

equilibrium, high ability students do avoid paying the effort cost c, which could be criticized on

normative grounds, i.e., perhaps the students are lazy. However, these students are not enrolling in

15

an ability-concealing department to hide their abilities, and this is the key point here. The small

percentage of low ability students who pool with the high ability students do so in order to hide

their (low ability) statuses; this group does end up with inflated grades, but the group is nonetheless

very small.

In the pooling equilibrium here, the ability-revealing department has no students in it. Presum-

ably this is not ideal for the department, and indeed one might conjecture that such a department

would anticipate a lack of students and change its grading policy prior to student enrollment deci-

sions. Department grading policies are probably sufficiently sticky so that changing grading norms

is not a simple process, and from this perspective treating department grading policies as exoge-

nous seems natural. Nonetheless, we are exploring the matter of strategic department grading

policies in other research.

Separating equilibrium. The model always has a separating equilibrium in which low ability

students attend the ability-concealing department and high ability students, the ability-revealing

department. In this equilibrium, whose existence is not a function of the relationship between

πH and c, departments do not contains mixtures of low and high ability students. Rather, in the

equilibrium all low ability students attend one department and all high ability students, the other.

This feature of the separating equilibrium has one rather notable consequence: grades appear

inflated in both departments. To be precise, in the separating equilibrium all high ability students

receive top grades—because they are in fact of high ability and are enrolled in an ability-revealing

department—and low ability students receive top grades, too—because they enroll in an ability-

concealing department which provides everyone with high grades. Thus, our model’s separating

equilibrium, which exists for all values of πH and c, features a distribution of grades that looks on

the surface to be highly inflated.

In fact, the distribution of grades in the separating equilibrium has literally zero variance be-

cause in it every student receives an “A.” These numerous “A” grades, however, reflect fundamen-

tally different dynamics. “A” grades received by high-ability students are accurate evidence of

16

excellent students being willing to subject themselves to an ability-revealing process; thus, these

“A” grades do not reflect grade inflation. In contrast, however, “A” marks received by low-ability

students are evidence of low-ability students avoiding an ability-revealing process; “A” grades re-

ceived by these students do reflect inflation, and thus the fraction of inflated grades in the separating

equilibrium is 1− πH .

The model’s separating equilibrium is more compelling than the previously-discussed pooling

equilibrium because the latter only exists when the fraction of high ability students is very large.

With this in mind, we argue that our model shows that student sorting by itself is sufficient to

lead to a situation in which all students receive identical grades, all of which are “A” marks; this

situation looks like one in which grade inflation is a serious problem but, at least for high ability

students, it is not.

We earlier mentioned that our assumption about the existence of only two departments in a

university is not binding. If there were more than two departments in our hypothetical university—

some ability-concealing and others ability-revealing—the separating equilibrium we have de-

scribed here would continue to exist as long as the ability-revealing department or departments

imposed effort costs beyond those imposed by the ability-concealing departments. The key to the

equilibrium is not the number of departments per se; rather, the key is the fact that ability-revealing

departments impose more of a cost on enrolled students than do ability-concealing departments.

Another notable feature of the separating equilibrium is that it requires only that πH be neither

zero nor one. If πH were one, then all students would be of high ability and the only equilibrium

that would exist in the model would be one in which students pooled on the ability-concealing

department. Observationally speaking, grades would appear to be inflated in this scenario, but in

reality they would not be because all students would be of high ability. If on the other hand πH

were zero, then all students would again pool on the ability-concealing department. This would

yield a situation with rampant grade inflation, one wherein all students of low ability are labeled

by an ability-concealing department as high ability.

17

Extension: continuous student ability and an education bonus

One might argue that our characterization of student ability as binary—either low or high—is too

coarse and that this may be responsible for the result, above, that, when students separate, there

is no variance in student grades. With this in mind, we now offer an extension of our model that

allows us to explore the consequences of allowing student ability to exist on a continuum. Along

with this change we also include in the extension an education bonus that a student receives if she

enrolls in an ability-revealing department. As shown below, the extension of our model does lead

to variance in student grades; however, it does not change our fundamental results about grade

inflation and the effects of student sorting on the distribution of grades.

Let student ability be denoted θ. We assume that θ ∼ U (0, 1), but this assumption is not

qualitatively necessary for the results that follow. We make the uniformity assumption because it

allows for a closed-form equilibrium characterization.

When student ability exists on a continuum, we can no longer speak simply of “low” and

“high” ability students. In addition, with continuous student ability we need a more refined char-

acterization of grades and of post-education wages. We continue to assume that an ability-revealing

department is one that assigns grades based on underlying student abilities, and with a continuous

distribution of abilities this is obviously a bit of an abstraction insofar as a finite number of class

letter grades—“A,” “A-”, “B+,” and so forth—cannot map one-to-one to a continuous range of stu-

dent abilities. However, one can imagine that an ability-revealing department issues class grades,

returns discriminating assignments, generates ability-revealing letters of recommendation, and so

forth, in such a way that a student who enrolls in such a department has an overall record from her

educational experience that perfectly reveals her ability level. Thus, in the extension of the model,

a student with ability θ who attends an ability-revealing department receives a grade of θ.

With respect to ability-concealing departments, we continue to assume that such a department

awards very high grades to all of its students. In particular, in the model extension we assume

18

that every student in it receives a grade of one. This is parallel to our earlier assumption that

ability-concealing departments award grades of “A” to their students.

In terms of wages, suppose that a student of ability θ whose ability is known to the labor market

receives a base wage of θ in the post-education market. In other words, the more ability a student

has, the greater the student’s wage will be, assuming that the market knows the student’s ability. If

in addition this student attended an ability-revealing department, then she receives a wage boost of

e ≥ 0. Such an education bonus is intended to capture the fact that studying in an ability-revealing

department and bearing the requisite effort cost can lead to increased knowledge and, accordingly,

higher wages.

As was the case in our initial model formulation, a student whose wage is not known receives

a base wage in the labor market corresponding to expected ability level where this expectation is

taken given equilibrium student behavior. Such a student cannot receive the education bonus e

because the only way to receive such a bonus is to enroll in an ability-revealing department, an

action that would signal the student’s ability.

Considering both effort cost, education, and the effect of ability on what we are calling base

wages, if a student with ability θ enrolls in an ability-revealing department, her net utility is θ−c+e.

Similarly, if students with θ ∈ Θ ⊂ (0, 1) enroll in an ability-concealing department, then each

student with θ ∈ Θ receives net utility of∫

Θθ dθ.

The equilibrium of the model extension depends on a cutpoint that we call θ̄, and this cutpoint

is characterized by student indifference between enrolling in an ability-concealing and an ability-

revealing department. The equilibrium is described in Lemma 2, whose proof is in the appendix.

Lemma 2. Let θ̄ = 2 (c− e).

If θ̄ ∈ (0, 1), then the equilibrium of the model is semi-pooling. Namely, students with ability

θ < θ̄ enroll in an ability-concealing department and students with θ > θ̄ enroll in the ability-

revealing department. Students with θ = θ̄ are indifferent between the two departments, and we

ignore these students since they are of zero measure.

19

If θ̄ > 1, then the equilibrium of the model is pooling wherein all students enroll in an ability-

concealing department. If θ̄ < 0, then the equilibrium of the model is pooling wherein all students

enroll in an ability-revealing department.

The knife-edge condition θ̄ = 1 (equivalent to c − e = 1/2) is consistent with either semi-

pooling as described above or students pooling on the ability-concealing department; without loss

of generality, we assume that θ̄ = 1 leads to the latter type of pooling. Similarly, the knife-edge

condition θ̄ = 0 (equivalent to c = e) is consistent with either semi-pooling or student pooling

on the ability-revealing department; without loss of generality, we assume that θ̄ = 0 leads to the

latter type of pooling.

Lemma 2 shows that the indifference cutpoint θ̄, which determines whether the extended

model’s equilibrium is semi-pooling (θ̄ ∈ (0, 1)) or is pooling (θ̄ ≤ 0 or θ̄ ≥ 1), depends on the re-

lationship between the effort cost c required of students enrolled in an ability-revealing department

and the education bonus e. We earlier argued that c should be thought of as small compared to

one because this parameter represents a short-term effort cost as opposed to a discounted stream of

wages. Therefore, we assume that c− e < 1/2; this implies that the long-term benefit of education

exceeds by a large amount the short-term costs that a student faces when taking a large number of

university tests. This assumption is intuitive, and thus we focus our attention on two possibilities

for our cutpoint, either θ̄ ≤ 0 or θ̄ ∈ (0, 1).11 These regions are depicted visually in Figure 2.

Suppose first that θ̄ ≤ 0. Because θ̄ = 2 (c− e), it follows that the education bonus e must be

greater than or equal to the cost c that students must pay when taking classes in an ability-revealing

department. When e ≥ c, all students attend the ability-revealing department and, accordingly, all

students’ grades are accurate measures of their abilities. There is no grade inflation when θ̄ ≤ 0

because, in this situation, there are no students in the ability-concealing department.

Suppose one were to argue on normative grounds that student pooling on the ability-revealing

department is a good thing, i.e., that society benefits when θ̄ ≤ 0. One might then ask, how might

11If the effort cost c is close to zero, then c− e will be small as well, even if e is negligible.

20

Figure 2: Extended model equilibria as a function of c and e

e

c

c = e

c = 12

+ e

Pooling

Semi-pooling

students be induced to pool in this way? Based on Lemma 2, the answer here is simple: our hypo-

thetical university should increase its education bonus e until the value of education overwhelms

the cost of attending an ability-revealing department. As soon as this happens, then all students

become willing to attend said department, an outcome that, as we noted, has no grade inflation at

all. It is also true that making the effort cost c small will have the same effect.

Now consider the case c > e with the previously-noted proviso that the effort cost c remains

small. Intuitively, the inequality c > e implies that education is valuable to those students who

enroll in an ability-revealing department—but not overly so. If one thinks of education like Spence

(1973), solely in the sense of signaling, then e = 0, in which case the restriction c > e certainly

holds.

When c > e the model’s equilibrium is semi-pooling, and the first feature to notice about

the equilibrium is that the fraction of students who enroll in the ability-revealing department is

1 − θ̄ = 1 − 2 (c− e). This expression is decreasing in the cost term c, which is intuitive: the

more effort required in the ability-revealing department, the fewer the number of students who are

21

willing to enroll in it. It is also increasing in e: the greater the value of education for those students

who work hard, the greater the number of students who are willing to enroll in an ability-revealing

department.

In the semi-pooling equilibrium, the average student grade is

∫ θ̄

0

dθ +

∫ 1

θ̄

θ dθ = 2 (c− e) +1− 4 (c− e)2

2

Differentiating this expression with respect to c yields 2 − 4 (c− e), which is positive as long

as c − e < 1/2. Thus, the more difficult the ability-revealing department becomes, the higher

are average grades in equilibrium, ceteris paribus. On the other hand, the opposite result hold

with respect to the education bonus e: the greater this bonus, the lower are average grades in

equilibrium, ceteris paribus.

The result about the effect of the effort cost c on average grades is particularly notable, and

there are two reasons that increases in c lead to higher grades on average. First, when the difficulty

parameter c increases, the indifference cutpoint θ̄ increases as well. When c is large, that is, only

high ability students remain in the ability-revealing department; when their abilities are revealed,

they are accurately revealed to be high. This is not grade inflation; to the contrary, it is accurate

reporting. Second, and on the other hand, an increase in c causes some students to select out of

the ability-revealing department and into the ability-concealing department; these students receive

high grades on account of enrolling in the letter, indeed grades higher than they should based on

their underlying abilities. This is a form of grade inflation, and the fraction of students with inflated

grades is 2 (c− e). Overall, the point here is that the increase in grades due to increasing effort cost

c has two components which manifest themselves similarly but have different underlying causes,

only one of which can be considered problematic.

When the education bonus e increases, then the effects described above move in the opposite

direction. In particular, an increase in e leads students who previously would have enrolled in the

22

ability-concealing department to enroll in the ability-revealing department instead. This has the

feature of both lowering average grades and decreasing the number of students whose grades are

inflated.

One can imagine a dean or other administrative figure in a university arguing in the face of

high grades that departments need to make their classes more difficult so as to drive down os-

tensibly inflated grades. In the framework of our extended model, this would be equivalent to

making c greater. Positing that increasing difficulty will lower average grades is, however, a partial

equilibrium assertion. Since students sort themselves conditional on department difficulty, making

an ability-revealing department more difficult will drive students away from it and thus have the

opposite effect of what the dean or other figure intended.

Such an argument holds conditional on the existence in our hypothetical university of an ability-

concealing department, and this highlights the possibility that there is a collective action problem

in university grading, one that the dean could overcome if she could simultaneously convince an

ability-revealing department to be more difficult while convincing (or compelling?) an ability-

concealing department to become ability-revealing. We will return to this point later. At the

moment, though, it suffices to note that, when students have an ability-concealing department as

an option, the more difficult the ability-revealing department, the higher are average grades and

thus the more grades look like they are inflated.

One seems a similar point when examining the variance in grades in the extended model’s

semi-pooling equilibrium. This variance is

∫ θ̄

0

dθ +

∫ 1

θ̄

θ2 dθ −

(2 (c− e) +

1− 4 (c− e)2

2

)2

.

Algebra shows that this variance approaches 1/12, the variance of the uniform distribution on the

unit interval, as c approaches e. This is appropriate because, as c becomes smaller, more students

enroll in the ability-revealing department and this leads to a grade distribution that gets close to the

23

true ability distribution.

More importantly, the derivative of the above variance is negative for relevant parameter val-

ues. In other words, the more difficult the ability-revealing department becomes, the less variability

there is in student grades. This is because increasing difficulty leads to an increasing number of

students in an ability-concealing department and accordingly less grade variability. If, say, our

aforementioned university dean or administrator were to argue that his or her institution should

seek extensive variability in grades—because, say, variability in grades makes it easier to distin-

guish low and high ability students—the implication of our extended model is that the dean should

insist that the ability-revealing department be as easy as possible on its students.

Given the definition of θ̄, another option open for a dean who wanted to increase grade variance

would be to encourage the ability-revealing department to increase its education bonus. This would

have the effect of inducing more students to enroll in said department, and this would lead to

increased variance in grades and less grade inflation. This option may be more palatable than

decreasing the effort cost c.

We motivated the extension of our model with the recognition that the coarse way in which we

modeled student ability—low versus high—diminishes our ability to ascertain whether in equilib-

rium there is variance in student grades. Our extension shows that this concern was indeed valid.

Namely, as long as the effort cost c associated with an ability-revealing department is not excessive

compared to the education bonus (that is, c−e < 1/2, which we earlier justified by arguing that the

short-term cost c is close to zero), then there is indeed variance in student grades. This variance is

conditional on effort cost c and the education bonus e, and above we have explained how changing

these two parameters changes average grades and grade variance.

Our final point about the model extension concerns the possibility of including a cost term for

enrolling in an ability-concealing department. Were we to have done this, then we would have

seen that what we call c in the model extension proxies for the difference in cost associated with

an ability-revealing department and an ability-concealing department. Having said this, there are

24

two ways to interpret the effects of an increase in the effort cost c. In particular, c in the extended

model can increase because the ability-revealing department becomes more difficult; and, it can

increase because the ability-concealing departments becomes easier. These dual perspectives on c

do not change any of our earlier interpretations or derivatives, but they do imply, for example, that

increasing the cost required for a student to enroll in an ability-revealing department is equivalent

to decreasing the cost associated with an ability-concealing department.

Discussion

We have offered a game-theoretic analysis of student grading, an analysis motivated by empirical

studies documenting upward trends in grades in higher education institutions. Our formalizations

shed light on the implications of student choice over departments, a key feature of grading pro-

cesses in universities that is often neglected in discussions of grading trends in higher education.

The students in our model are both effort-averse and forward-thinking, and this induces a dynamic

in which the best students seek to distinguish themselves from their lower-ability counterparts and

are willing to undertake costly behaviors so that their true abilities are revealed to a post-education

labor market. The end result of this is that average grades are high but not because of, say, lax

standards or enrollment pressures. Rather, grades are high because good students appropriately

earn them from an ability-revealing department and lesser students garner them, so to speak, from

an ability-concealing department.

To be clear, we are not arguing that our model should be thought of as a (or “the”) comprehen-

sive explanation of grade inflation in higher education. Our primary objective has not in fact been

to offer a complete theory of grading in educational institutions but rather to encourage scholars

interested in grading to consider assiduously the consequences of student sorting on grade distri-

butions. Existing literature makes it clear that there are a variety of explanations for the types of

grade inflation that empirically-driven scholars of education have identified, and our models should

25

remind those considering these explanations to be mindful of how sorting can manifest itself.

The model adduced here shows how student sorting at one level of the higher education

landscape—students within universities—leads to distributions of grades that seem extensively in-

flated but, for high-ability students at least, are nonetheless accurate. There are additional levels of

sorting that we have not specifically engaged, and these include sorting across universities and sort-

ing within departments. Although we have not modeled choice of university, one could envision

a more general model of education wherein a student selects into the best university that accepts

him or her and then chooses a department within the chosen universities. If some universities are

known for being ability-concealing and requiring little effort, then students of higher ability will

presumably not apply to these institutions and instead pursue education in costly, ability-revealing

institutions. These latter institutions will then be disproportionately populated by high ability

students, which will compound the grading dilemmas caused by within-university selection into

departments. Given the recent increase in competition in the United States competition for admis-

sion into elite colleges and universities, it is conceivable that across-institution sorting may be a

notable factor in explaining nationwide increases in average grades.12

What does our model say about contemporary trends in grades? Perhaps the most direct im-

plication is as follows: within-institution trends in grades are hard, if not outright impossible, to

interpret in isolation from trends in the extent to which students sort themselves strategically into

departments. Put another way, student sorting confounds grading, and therefore analyses of grade

trends that are executed independently of sorting dynamics can be misleading. In our basic model

where parameters are reasonable (i.e., the cost of attending an ability-revealing department is not

too high compared to long-term wage streams), literally all students receive high grades and there

12Not all schools may be subject to sorting in the same way; the University of Texas, for example, issubject to a state law that mandates admissions policies for freshman from Texas high schools. This lawmay lead to a larger variance in underlying student ability levels than would be expected otherwise. See“Report to the Governor, the Lieutenant Governor, and the Speaker of the House of Representatives on theImplementation of SB 175, 81st Legislature For the period ending Fall 2014,” The University of Texas atAustin, available at https://www.utexas.edu/student/admissions/research/SB_175_Report_for_2014.pdf (last accessed May 12, 2015).

26

is correspondingly no variance in the distribution of grades. Viewed from the lens of grades only,

this situation looks problematic and in need of remedy; it is problematic, however, only for low-

ability students as the high grades received by top students correctly reflect these students’ high

ability levels. Challenged to defend its plethora of high grades, an ability-revealing department in

this situation might respond, “All of our students are excellent!” Due to student sorting driven by

high ability students seeking to distinguish themselves from low-ability students, this claim would

be accurate.

Put another way, our models show that the grading practices of individual departments cannot

be assessed simply by observing whether they assign many “A” marks or, say, “C” marks. Suppose

that a university dean were to compare the grades across two departments in her jurisdiction, and

suppose that she were to notice that both consistently give many (or perhaps exclusively) “A”

grades. Should the dean insist that these two departments raise their grading standards or, say,

ramp up the effort levels required for classes in said departments? Not necessarily. Of the two

departments, if one is ability-concealing, then only top students choose the other, thus inducing

this department to give a plethora of high grades that are accurate. If this department were to raise

its standard, this would not alleviate the selection incentives that we have explored here.

Our model is not explicitly dynamic, but it nonetheless suggests that one source for observed

trends in grades could be the emergence of one or two ability-concealing departments in a univer-

sity. That is, suppose that many years ago all departments in a hypothetical university were ability-

revealing and entailed effort costs. Were this the case, then we would expect these departments

to have issued both low and high grades. Suppose then that an exogenous shock—the Vietnam

War, as some have conjectured13—led one department in a university, or perhaps a small number

of departments, to adopt grade-inflating practices and simultaneously reduce the effort needed to

enroll in said department or departments. As soon as this were to have happened, we would be

13For example, see “A History of College Grade Inflation,” The New York Times, July14, 2011, available at http://economix.blogs.nytimes.com/2011/07/14/the-history-of-college-grade-inflation/ (last accessed February 13, 2015).

27

in a situation where the university had a combination of both ability-concealing departments and

ability-revealing departments. In the presence of both types of departments, our model suggests

that forward-looking students of high ability will seek to separate themselves from lower-ability

students, the latter of whom will choose ability-concealing department and the former, ability-

revealing departments. The result of this will be that all students earn high grades. In this example,

the culprit for high average grades overall is the presence in a university of a small number (or

even a non-small number) of ability-concealing departments. Indeed, one could argue that ability-

revealing departments are somewhat at the mercy of ability-concealing departments: once some of

the latter exist, the former will enroll only good students. This leads to a flattening of the grade

distributions produced by grade-revealing departments.

This point highlights a collective action problem associated with grading. A department that by

itself wanted to address institution-wide grade inflation can be stymied by the ability-concealing

behaviors of other departments. If an ability-revealing department were to make its classes increas-

ingly challenging in an attempt to mitigate inflation, then it would make the overall grade inflation

problem worse and in so doing decrease its own enrollments. To the extent that low enrollments

are problematic for departments who might want to use use enrollment figures to argue for faculty

positions, no department has an incentive on its own to increase the cost associated with its classes.

This sort of collective action dilemma means that university administrators should not assume that

individual departments will ever be able to coordinate themselves and form a solution to what

administrators might consider a grade inflation problem.

Our model has implications for department enrollment patterns associated with gender, race,

and socioeconomic status. We have assumed throughout this paper that students know their own

ability levels, but it worth considering how it is that a student might learn whether she is of low or

high ability and whether it is possible for a student to think, wrongly, that she is of low ability when

she actually is of high ability. This sort of error is particularly pernicious for students because

a high ability student who believes that she is of low ability may prefer an ability-concealing

28

department over an ability-revealing department—even though the latter would be more valuable.

Moreover, if pre-university evaluations for a given group of students are systematically biased

in this way so that students in said group regularly underestimate their abilities, then this group

will enroll in ability-concealing department even though they should not. This will lead to labor

market inefficiencies (high ability individuals will not be treated as such) and depress the group’s

aggregate, post-education earnings in the long run.

There is evidence that these kinds of concerns shape student enrollment patterns across depart-

ments. Ehrlinger and Dunning (2003) show that women often underestimate their scientific knowl-

edge and aptitude. According to our model, this will lead women disproportionately to select into

ability-concealing departments, and this will have society-wide implications, namely, potentially

wasted talent and a lack of women in certain scientific fields. Similar concerns have been raised

about African-American students (Nacoste, 1989; Ewing et al., 1996), and there is evidence that,

ceteris paribus, students of low socioeconomic status are less likely to pursue further academic

study than students of higher socioeconomic status (Erikson et al., 2005; Jackson et al., 2007). Per

our model, the culprit for this could be systematic biases in pre-university grades which lead low

socioeconomic students to doubt their own abilities.

A final point on this subject concerns risk aversion. If a group of students is more risk averse

than another, this could lead members of the group to enroll in ability-concealing departments

rather than risking enrolling in the ability-revealing departments. The relationship between gender

and risk aversion is not simple (e.g., Jianakoplos and Bernasek, 1998; Schubert et al., 1999; Eckel

and Grossman, 2008), but it is worth pointing out that, if women are more risk averse than men,

then this dynamic will compound the types of biases, noted above, that drive women dispropor-

tionately toward ability-concealing departments.

We conclude with comments that link our game-theoretic approach to empirical work on grade

inflation, and here we pursue two approaches. First, we discuss our model’s implications for the

ways in which one might seek to understand whether grades in a given institution are inflated. And

29

second, we consider how one might test to see if our model’s dynamics roughly approximate what

one observes in higher education institutions.

With respect to the first point, suppose that a researcher at a university wanted to know about the

extent to which her institution’s departments were ability-concealing or ability-revealing. Knowl-

edge of this type could in principle inform university policy decisions insofar as regulating ability-

concealing departments by compelling them to issue ability-revealing grades could “solve” the

problem of grade inflation to the extent that it is considered a problem. How would an interested

party determine which departments in a university are ability-concealing? As is clear from our the-

oretical results, looking at which departments have high grades is not sufficient and can actually be

misleading. Rather, ability-concealing departments can be identified because their students vary

in ability yet receive similar grades. Presumably the administration in most if not all educational

institutions knows, say, standardized test scores of all its enrolled students. Departments whose

enrolled majors, say, have high variance in test scores yet have received disproportionately high

grades may be ability-concealing.

To be precise, suppose that of two departments in a university, one had high variance in student

test scores and low variance in grades, and suppose that the second had low variance in test scores

and high variance in grades. This would presumably indicate differences in grading practices in

a way that falls roughly along the ability-concealing versus ability-revealing dichotomy described

here. These variances are still subject to a student sorting confound, but this is generically true

unless a university were to compel all of its students to enroll in common course. Absent such

a common course, combining variances in measures of ability like standardized tests with vari-

ances in grade distributions might yield a plausible picture of which departments in a university

are ability-concealing and which are ability-revealing. In the long run this will aid the general un-

derstanding of grading dynamics and how educators and researchers should interpret trends—both

upward and downward—in grades.

Another approach to assessing whether grades in a university are inflated is to consider whether

30

post-education institutions that in theory rely on these grades actually use them. Universities

know the enrollment choices of their students and which students receive high grades; univer-

sities will often also know which students, say, receive interview opportunities and valuable job

offers. Armed with this information, universities could in principle determine which grades are

correlated with post-education success. For example, if there is a department that produces high

grades which do not predict post-academic success, then it follows that this department may be

guilty of grade inflation. Similarly, if there is another department such that grades from it—low

or high—predict success, then this department is presumably ability-revealing and thus attracting

the best students. Moreover, if a university observes that its students are subjected to grueling and

extensive interviews, this might suggest that none of its grades is particularly meaningful. Long

interviews are only necessary, one would think, if grades are not signaling abilities, i.e., if there

is university-wide grade inflation. Similarly, if student activities like internships and formal rec-

ommendations carry more weight in the post-education labor market than grades, then one might

surmise that grades are lacking in their signaling ability.

With respect to our second empirical point, we now consider four empirically testable hypothe-

ses based on our model.

First, students will sort themselves based on self-perceived ability across types of departments.

This hypothesis could be tested by conducting a survey among undergraduate students prior to

enrollment decisions. Our model suggests that student perceptions about the difficulty of grading

in a department should be correlated with the perceived ability level of the students majoring or

concentrating in that department. Students who perceive themselves as high ability should cluster

in difficult concentrations, ceteris paribus, while students who believe they are low ability will do

the opposite. Since an important implication of our model is that the grade distributions within

a department cannot be, on their own, evidence of grade inflation, clustering in self-perceived

student ability could be an important sign that grade inflation is occurring at disparate rates among

departments.

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Second, employers’ preferences for hiring students in departments perceived as ability-

revealing will stem in part from the perceived higher ability level of students in those departments.

This hypothesis could be tested using a survey of employers who frequently recruit undergraduates

on a particular campus. Employers could be asked about whether they prefer certain majors, how

they interpret grades among students from different majors, and how they perceive the grading

practices within those majors. Employer reactions to mock student profiles that only vary based

on the major granting department could be used to measure how much of a difference perceptions

of different majors make. These responses could then be used to assess the weights employers

place on grades varies with the perceived grade inflation in the major. In the survey considered

here, it would be important to focus on comparing departments that have similar course content

and employers that do not have a strong preference for students with specific technical skills.

Third, departments which place a high value on attracting the highest ability students will

be the least likely to have inflated grades. A key implication of our model is that high ability

students will select into non-grade inflating departments. Consequently, departments that want

to attract these students will have an incentive to provide grades that accurately reflect student

abilities. This hypothesis could be tested by conducting a survey of department faculty regarding

both the types of students a department is seeking to attract and the department’s grading policies,

whether it mandates curves, mandates median grades, and considers grade distributions in the

context of course evaluations. We would expect departments that put an emphasis on attracting

the highest ability students to be the most aggressive in combating grade inflation, ceteris parbus,

while departments that emphasize enrollment maximization regardless of student ability would be

more tolerant of grade inflation.

Fourth, student sorting will increase in the presence of increased information about department

grading practices. Students can only sort themselves across department types if they know which

departments are ability-concealing and which are not. This said, it is probably rare that students

have perfect knowledge about department grading practices. Consequently, in situations where

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information about department grading practices suddenly increases, we would expect a surge in

student sorting. Suppose that an institution were to promulgate a policy by which class median

grades are made public. Our model suggests that this sort of natural experiment will lead to in-

creased student sorting.

With this last point in mind, we end by noting a previously-cited study that draws on data

from Cornell University. As we described earlier, Cornell began publishing information about

course median grades in 1996, and Bar, Kadiyali and Zussman (2009) discuss changes in student

enrollment patterns in Cornell following the 1996 policy change. In particular, they show that

enrollment in low grading classes declined after Cornell began publishing median grades; these

classes are presumably ability-revealing. However—and this is the key point—this effect was large

for students with relatively low grade point averages and yet statistically indistinguishable from

zero for students in the top ten percent of Cornell’s grade distribution. In other words, Cornell’s

policy change, which allowed students to determine which classes were ability-revealing and which

were ability-concealing, produced a situation wherein low ability student shifted out of the former

in favor of the latter. High ability students did not behave this way, and this is what our model

would suggest. That the Cornell study produces results in line with our formalization is pleasing.

This emphasizes the role that student sorting plays in grading and that incentives for sorting should

be a part of the academic literature on grading practices.

Appendix

Proof of Lemma 1.

Proof. Suppose that πH ≥ 1 − c. If all students enroll in a grade-inflating department, then each

receives πH as a post-education wage. This situation obtains because the labor market cannot

distinguish the students based on their transcripts, all of which contain identical marks (all “A”

grades, as the ability-concealing department by construction issues only “A” marks) and all of

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which were issued by one department. A low ability student who deviates to a grade-revealing

department will pay a cost of c and, by virtue of enrolling in an ability-revealing department, will

be identified as having low ability. Such a deviating student will thus receive utility of −c, which

is strictly less than πH . In contrast, a high ability student who deviates to the ability-revealing

department will receive a wage of one, which is better than receiving πH , but will also have to pay

a cost of c. If πH ≥ 1− c, then this deviation is not optimal.

Now let πH and c be given such that πH ∈ (0, 1) and c ∈ (0, 1), and consider the existence

of a separating equilibrium as posited in the lemma. In such an equilibrium, low ability students

receive utility of zero. If a low ability student were to deviate to the ability-revealing department,

she would continue to receive a wage of zero and in additional would pay the effort cost c. Since

c > 0, deviating in this way is not optimal. Similarly, with separation a high ability student receives

a wage of one and an overall utility of 1− c. Deviating to an ability-concealing department would

lead to a utility of zero because the student would be considered low ability. Since 0 < 1 − c,

deviating is not optimal.

To show that there is no pooling equilibrium in which all students enroll in a grade-revealing

department, suppose that such an equilibrium were to exist. A low-ability student would earn

utility of −c < 0 in such an equilibrium. Were such a student to deviate to an ability-concealing

department, the student would not pay the effort cost c and would receive a wage between zero

and one, inclusive. This would lead to positive utility greater than −c, thus contradicting the

conjectured equilibrium.14

As an aside, this proof does not depend on the value of c being identical for both low and high

ability students. This point supports the claim made earlier in the body of the paper that, as long

as enrolling in an ability-revealing department imposes an effort cost, the results of the model do

14We need not specify the off-equilibrium path belief of the labor market that a deviating student inan ability-concealing department is of high ability conditional on all students pooling on enrolling in anability-revealing department. This is because any such conditional belief will contradict the existence of theconjectured pooling equilibrium.

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not depend on whether c varies by student ability.

Proof of Lemma 2.

Proof. If in equilibrium a student with ability θ1 prefers an ability-revealing department, then it is

straightforward to show that a student with ability θ2 > θ1 does as well. This follows from the fact

that a student’s utility for an ability-revealing department is increasing in underlying student ability.

Similarly, if in equilibrium a student with ability θ1 prefers an ability-concealing department, then

it is straightforward to show that a student with ability θ2 < θ1 does as well. Thus, if in equilibrium

students separate in their choices between departments, they must do so where a cutpoint, which

we denote θ̄, partitions students based on ability.

We now derive the value of θ̄. Let a student with ability θ = θ̄ be indifferent between enrolling

in an ability-concealing department and enrolling in an ability-revealing department. The former

nets the student θ̄/2 and the latter, θ̄ − c+ e. Equating these two utilities yields θ̄ = 2 (c− e).

If c− e ≥ 1/2, then θ̄ ≥ 1 and all students pool on the grade-revealing department. Similarly,

if c ≤ e, then θ̄ ≤ 0 and all students pool on the ability-revealing department. If θ̄ ∈ (0, 1), the

equilibrium is semi-pooling.

35

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